Rationally speaking, is a presentation of New York City skeptics dedicated to promoting critical thinking, skeptical inquiry and science education. For more information, please visit us at NYC Skeptic's Doug. Welcome to, rationally speaking, the podcast, where we explore the borderlands between reason and nonsense. I am your host penalty and with me, as always, is my co-host, Julia Gillard. Julia, where are we going to talk about today?

Masimo, this is the second of two episodes that we're taping with Graham Priest, who is the Boyce Gibson professor of philosophy at the University of Melbourne and a distinguished professor of philosophy at CUNY Graduate Center in New York. Our last episode with Graham, you may remember, was about Eastern philosophy. And this time we're going to talk to him about another specialty of his, which is Parro consistent logic.

So, Graham, well, we'll delve into this in a minute.

But just just to offer a teaser, it may seem like incredibly self-evident, like like one of the most self-evident truths there could possibly be that something can't be simultaneously true and false.

Well, that that is a self-evident truth. We are going to call into question that.

So, Graham, could you give our listeners just a brief definition of what power consistent logic is?

OK, if you go to a university and you take a first course in logic, you will almost certainly be taught that once you have a contradiction, everything follows from that, a contradiction being.

Well, let me give you an example, right. You will be taught that if you suppose that the sun isn't is not made of green cheese, it follows that I'm a frog.

Clearly obvious. Right. OK. And why is that? OK, let's let's not go into too technical.

OK, I'm sorry I did quickly 20, 30 seconds into Pakistan has withdrawn gone that that does seem kind of crazy.

OK, because the moon being or not being made of green cheese would seem to have nothing much to do with you or frogs.

OK, would seem. It would seem.

It would seem yeah.

But this this illustrates the fact that in in the logic education of the colonies receive, you will be taught that if you've got an inconsistency of the kind that I gave you, that absolutely everything follows, that you're a frog, you're not a frog, that the mood is purple, that Socrates was Chinese, that I'm in on the moon, you know, an explosion of nonsense.

Yes. Yes. So this principle of inference is often nowadays called explosion, perhaps because the inconsistency explodes right now that that principle of inference explosion is a relatively recent piece of logical technology. You don't find it in ancient Greek logic. It's debated in medieval logic. It doesn't really become orthodox in logic until about the turn of the 20th, the beginning of the twentieth century.

And is it the case that explosion is important because it shows you why you can't have contradictions.

Many, many people will say that explosion shows you that you can't have a contradiction, because if you did, then everything would follow. And that's crazy.

It's weird because it seems to me like you like the reason you can't have a contradiction is just because it doesn't make sense.

It almost seems like you don't need an explanation for why it doesn't make sense to contradict yours.

OK, that's a very traditional view. I mean, that was. I think. I think so. Let's come back to that in a second.

I mean, your original question was what is a yes? I'm sorry.

I'm sorry, but I want to explain what an explosive logic is. It's one where these contradictions explode a power. Consistent logic is exactly one where explosion is not valid.

So you can have a contradiction. Some things may follow from it, but it's not the case. Everything follows. So if you're dealing with this kind of logic, you can treat contradictions as kind of singularities. They're kind of isolated, so they don't sort of ruin things in a way. That explosion will ruin something, a theory that's inconsistent.

It's a way to contain the logical explosion. Basically, it's a way to contain contradictions and make sure they don't sort of infect other things.

So you can contain the contradiction that the moon isn't is made of green cheese. And it won't affect anything about you being a frog, for example. Will it affect anything else besides.

Yes, it kind of it depends on interconnections. So one reason why people are interested nowadays in inconsistent theories is because of certain paradoxes that arise in a series of sets of truths and so on. And if. You have general principles of success, then contradictions about some states will have other implications for some sects.

So there can be consequences, but the controlled consequences and not this kind of lunacy about everything, following what would be an example of a paradox that consistent logic was was intended to deal with.

Before I answer that question, let me say something about dilithium, dilithium is the view that some contradictions are true. OK, so it's not a view about logic, it's a bit about semantics or metaphysics, but obviously there's a connection because if you think that some contradictions are true. And you don't think everything is true, then you must subscribe to a consistent logic, because you've got this true contradiction, you don't want to explode in front of you. All right.

Maybe this is just my traditional self talking, but when you say some contradictions are true, I want to ask, well, which part of the contradiction is not all that can be true?

The contradiction, both parts I know will come back to this minute.

But let me just finish drawing this distinction, OK? Because he did ask me what sort of examples that were and, you know, I'll answer that.

So if you hold this metaphysical view totally theism, you'd better subscribe to a consistent logic. But there are lots of people who work on Parkinson's logic who are not Dyle artists because they just think the explosion is crazy.

So, for example, Parkinson's Logic has applications in information processing where data can be corrupt and contradictory and you don't want your data to explode. Or if you look at the history of science, it's pretty clear that there are actually explicitly inconsistent theories, such as the early calculus, such as opposed to the atom and these guys reading reasoned in a sensible fashion.

Now, I'm not suggesting that these theories were examples of dilators. I mean, Bolls theory was never, you know, accepted as literally true. But nonetheless, these guys had to use a consistent logic, whatever it was, enjoying their conclusions. OK, so because at that moment, let me let me see if I understand this correctly.

For instance, in the case of Bhau, I looked into that when when I read your article in the San Francisco Opinion Philosophy, which does mention that example. So you're saying that they had to be used more, for instance, that had to be using a consistent logic, whatever that was, because it was not resolved that he was using a theory that later turned out to be incorrect or rejected or whatever. But at the moment, he was using it to derive partially valid inferences.

And yet somehow part of the theories construct were, in fact, contradictory, right?

Yes. I mean, he was deploying the quantum principle, which says energy is discrete and he was using classical mechanics in which it's continuous. And he was making lots of interesting predictions about, you know, the spectrum of the hydrogen atoms, which turned out to be right. OK. The theory turned out to be wrong in the end. But I mean, Bob was well aware that he was using inconsistent theory. He wasn't very explicit about what was allowing him to draw inferences.

But it must have been I mean, obviously, he didn't, you know, conclude the spectrum. The hydrogen atom has 3407 six lines, which would have followed an explosion. So, I mean, he must have been deploying some kind of power, consistent logic. So that's just to say that many people would think that Perkins's logic is correct without being divests. OK, well, I'm entirely system because that's all ground clearing for your question. So what kinds of things what kind of contradictions might be true?

Well, several different things have been suggested, but I think probably most people's favorites are certain paradoxes. So the paradox of self reference, some of them are very old, so probably the oldest and most famous is something called the liar paradox. I'll tell you what that is at the moment.

And that was discovered about the third or fourth century BCE in Greece and some of these paradoxes of relatively recent. They turned up in a very unexpected place at the end of the 19th century in the foundations of mathematics. So one's immediate reaction to these things might be to write them off as a kind of party game. But you can't do that now just because of the central role that these things have played in the foundations of mathematics in the 20th century. OK, so so what sort of paradox is my talking about?

Well, the easiest one to get your head around is the liar paradox.

So suppose I say that these very words that I'm now uttering are false is what I say true or false? Well. Is it true? Mm hmm. Well, but what did I say? I said these very words, I'm uttering a false and if that's true, well, they're false. So if it's true, it's false. Right. OK, so suppose it's false. Well, what did I say?

I said these very words are false.

And if they are false, well, they're true. So they seem to be both true and false. Whichever way you go.

Can they be neither true or false? Can they be nonsense just like it? It's neither true or false that the ghost act to stem the doubters because the sentence is nonsense, you know? Well, I use Made-Up words, but I could make I can make a nonsense sentence with English words, too.

But the thing is, I'm sorry. Before I answer that question, there is a difference between literal nonsense, you know, like you can make up words or whatever. And what what you just said. I can understand what he said. But again, attach a meaning to what he said.

It's just that when I kind of nonsense. Well, that may be. But what I'm saying is there is a difference there because I can understand semantically what he's saying. And it's the problem is that when I try to make sense of the logical consistency of what he's saying, then I run into the paradox. But prima fascia, it's you know, at first glance that this is I understand perfectly well what he's saying.

Yeah, I you can say the words are meaningless. You can say they're neither true or false. These are not the same thing. Right. And I think the for the reasons that Masimo gave, the view that these words are meaningless isn't very plausible. Their grammatical and you understood well enough to follow the reasoning and so on.

The the suggestion that they're neither true or false, I think has has more legs than indeed, this is one of the standard views about these paradoxes, are there are problems with it. Let me explain one. OK, we're going to have to do a little bit of reasoning here, so we'll take this slowly.

OK, OK.

The paradox I gave you before was the liar paradox and the words were what I'm now saying is false.

OK, and let's consider the possibility that those words are neither true or false. That'll that'll do it for you. OK, because what we saw was if it's true, it's false. It's true. OK, so maybe neither. Problem solved. OK, but I want to consider something else. I'm going to say something else, which is this. What I'm now saying is either false or neither true or false. So I've taken your suggestion on board, right?

Mm hmm. And one I'm now saying is either false or neither true or false. OK, what's the status of these words? Well, if they're true. Then they're either false or neither true or false, and that's a contradiction. If they're false, well, if they're false, then they're certainly either false or neither true or false. The total sentence is true.

So true. OK. And now if you want to say, hey, Kareem, these things are neither true or false, then I say to you, well, Julia, if they're neither true or false, well, then they're certainly either false or neither true in office.

And so like that.

This is a phenomenon which logicians tend to call revenge, because revenge revenge is the revenge of the.

And it's a very common phenomenon in discussing paradoxes of this kind. Whenever a theorist comes along with a sort of putative solution, you can actually take the concepts that they deploy and you can kind of give an extended paradox or a revised paradox or revenge paradox.

The thing that these paradoxes all depend on self reference and self reference allows you to kind of take any division and twist it and destroy it and break it.

So self reference is kind of a construction which can be used to tear boundaries. And, you know, any solution is going to put up boundaries and then you deploy self reference to tear them down.

So, I mean, this is part of the argument. Why? Consistent solutions to the paradox of this kind ain't going to work because you've got machinery to tear through boundaries and the Dyle approach says, hey, you know, yeah, these things are both true and false.

This is a very strange creature which actually crosses boundaries and also permeates between boundaries, between true and false or whichever boundary you want to draw.

So when I suggested that maybe we should say the sentence is meaningless, um, what I what I sort of meant by that was that maybe we should declare the concept of self reference to be not to not make sense, but that you just can't refer to.

But they can do that because it referred to itself, but it actually does.

But the whole concept of self reference does do quite a bit of work in mathematics and logic and. Right. So you can't you really can't get rid of it, although I'm glad that you actually stated what I was about to ask it, which is it seems to me that a lot of these paradoxes, if not all of them, that do deal with self reference.

So does that is it is it reasonable for you to say, for instance, for somebody to say that, OK, look, then what we can do is to deploy, let's say, classical logic in any case, other than cases where there self sufferance and then when there is self preference, we do something else.

What I'm going at getting out here is that logic.

After all, one way to look at logic at least, is as a set of tools that have a proper domain of application, just like every tool. And so you run into an issue where that tool doesn't work. And instead of trying to, you know, use a screwdriver to hammer something in your nail and say, well, I need a hammer now, I need something else, does that is something like that available to the logician?

Well, the quick answer is yes.

OK. So is that it depends. The details depend on how you conceptualize logic. There are certain people who think that there's no such thing as the one true logic, that logic is at all very much like you described, and you can help yourself to any bit, which gives you the right result.

So you're kind of an instrumentalist about logic, and that's possible.

There are some people who think, no, no, no, there's one true logic. You can't be a pluralist. If you subscribe to that view, then you you certainly can't run the line. I've just described what you can do, something very similar. So if you. Look at the way that logic works then. Logic deals with reasoning about a whole bunch of cases. Mm hmm. Okay. And sometimes you will find yourself in a special case, in which case you can use the special case of you can consider the general logic which is restricted to that special case.

And then that might be that might give you the effect of using a different logic, although you haven't changed your logic, you're just considering the application to a special case.

So in either of those ways, if you are reasoning about a situation which is not really thick, then you're in this kind of special case and you can use standard logic on it.

First year logic, which has explosion. But if you are in a situation where which throws up these dilators, well, then you can't do that. You have to use the full strength of consistent logic.

So there are there different kinds of power, consistent logics, and if so, which one do you subscribe to?

Or what are some, like, distinctive features of the one that you subscribe to, right power, consistent logic is not one thing.

There are many, many. I mean, there's interesting history here. I mean, modern power objects were constructed for the first time in the 1960s, 1970s, and they were constructed in many different countries by people who are unaware of what people in the other countries were doing. So they were constructed in that.

It was before the Internet, I suppose. Yes.

So they were constructed in Brazil, in Poland and Canada and Australia. And often these logics are constructed on very different assumptions.

And it may well be that if you're using logic instrumentally, you would choose different kinds of logics, different kinds of Parkinson's, not X, but the generally speaking, the sort of consistent logic that I favor for discussions of things like the liar paradox is one kind of consistent logic called a relevant logic, which was actually invented by a couple of American magicians, Anderson and Belnap, in the 1960s. Mm hmm.

And what about the Belnap insisted was that if you've got an inference, then the premises really must be relevant. The conclusion that sounds kind of strange, right? But it's it's a very obviously. But if you enforce this, it's obviously going to rule out things like explosion because, you know, in a in an instance of explosion, the contradiction is obviously relevant to an arbitrary conclusion.

So is there any more formal way to define what's relevant or. Yes, there is. OK, if I'm not sure we want to go into that.

OK, well, I'll leave it to you.

Well, OK. To do it properly, you need to go into the technical details and I think probably that's not an appropriate thing for this occasion.

OK, but I do want to follow up on what you were discussing a minute ago about let me go back to this idea of logic as either universal or instrumentalist that these these two views are logical.

So one of the shocking or so revealing turning points in my in my academic career was when I went back from being a regional biologist, went back to graduate school and started taking courses in philosophy. And one of the courses that one of the courses in the book that came came with it was Logic's plural. And I looked at it and I said, what does something strange here? This this could could it be that they made a mistake in spelling on the on the cover of the book?

And no, it turns out it wasn't. I was introduced to this idea that there are different kinds of ideas that can be deployed at different domains of application and different properties.

Now, that said, however, could it be that there is a reasonable position that let's let's say I would I could define for lack of better terms as a quasi instrumentalist? Because in this sense, yes, different logics have different domains of application, different properties.

And in fact, your own example of people developing different kinds of production systems, extending from different axioms, different assumptions, fits the bill. But it's not that you can do that arbitrarily. So it's now you can just invent whatever the heck you want and make sense of it in a similar sense. It seems to me to, let's say, mathematics. You know, you can have your geometry, right? You can have a Euclidean geometry, non Euclidean geometry.

They are they do start from different set of assumptions. They have different consequences and properties in different domains of application. But it's not that you can say, therefore, anything goes and I can just invent my own kind of geometry and he's going to he's going to do any arbitrary work. Right. So in that sense, one could still salvage this idea that logic is not arbitrary and therefore, in some sense universal, like like math would be universal.

But but that doesn't mean that it's not pluralistic.

And that's quite a coherent view if you think that the kind of a police escort is under my rubric, because I just been told by a logician that I have to go ahead.

Well, but if you had an incoherent view, I'd be fine right now, of course, to play cards and logic. Hang on. Hang on. You shouldn't identify contradiction with incoherence, right?

Explosion turns a contradictory situation into an inka one. Good point. That's interesting. Yeah. Which reminds me, we were going to come back and talk about why you might think the contradictions can't possibly be true.

But let's start with mathematics first.

So suppose you hold the view that different domains of inquiry or different topics require different logics. You may well hold the view that the domain of.

Topic nonetheless requires that the right logic for that to mean off topic. OK. All right. And that's some kind of pluralism. And, you know, there are some Perkins's magicians who hold that view. I think Newton DaCosta, the one of the first Parkinson's magicians who invented a number of Parkinson's systems in Brazil. This was his view. I think this is his view. He's still alive.

But I think one needs to be careful here because there's there's an important distinction that often gets split over in discussion of logical pluralism are in one sense, logical. Pluralism is obviously true. There are many different logics, like there are many different geometries.

OK, you know, this classical logic, that's the logic of Frager and Russell invented 100 years ago. Those Aristotelian logic, those are consistent logic with intuitionist logic.

I mean, there's obviously a plurality of these things. But then, of course, there's a plurality of physics as well. You know, this Newtonian physics, there's there's quantum physics, there's Aristotelian physics.

That is a plurality of theories.

Doesn't tell you that, you know, there is one one. Right, one. Right. OK, similarly, the fact that there's this plurality of logics doesn't tell you that there isn't one right one.

So, in fact, if I may help you on this one for for the benefit of our listeners, the example when Newtonian mechanics is particularly interesting, because, of course, the first reaction that could be. Well, yes, but we know the Newtonian mechanics is wrong and, you know, said let's say relativity is right. Yeah, well, maybe, except that we do use Newtonian mechanics for all sorts of actual practical purposes, including sending space probes around.

So it is actually useful, even though we know that it's not technically correct. Absolutely.

And I mean, this brings us back to our discussion of whether you can use different objects with apps without suggesting there's one true logic. Right. Because Newtonian mechanics is a sort of special case restricted case of relativity. Not not exactly.

But, you know, for low velocities, the practically the same.

Right. OK, so you can use Newtonian mechanics because it is you're dealing with a special case of low velocity in a similar way. You might subscribe to a Parkinson's logic, but then in the special case where you're not dealing with these inconsistencies, you can use classical well, you can use classical logic because it's it's practically the same.

It seems to me at least like it would also be plausible that you that there wouldn't be maybe one true logic, but that you still have to just pick one like you can pick anyone you want, but then you have to stick to that. That seems like another plausible view.

Would you say the same is true in physics? No, no. Because logic is like a creation that we it's something that we've created. And and the form that it takes depends on like the starting axioms or how we want to define it. And that's not true.

Physics or. Yes, it is. I mean, Newtonian. Well, it's your theory of how the world works right now. That's true. Yeah. So so both physics and logic, you can have theories. Okay.

Now, what I mean, if you're a realist in physics, you suppose that all these different theories can't all be true, one must be right now, why should you suppose that logic is any different?

You've got lots of theories. Well, what would you what would you hold the theory up against to see how how well it does the way we do with scientific theories, the facts.

Logical facts, mathematical facts, yes, right, but don't certain of what are logical facts seem to depend on what logic you choose?

So no, no, I understand. I consider grammar. I mean, you know, technical grammar in the sense of, you know, Chomsky's grammar or something like this. OK, when you've got a language such as English, you can construct different grammars.

Some of them are right. Mostly they're wrong. But the aim of one aim of linguistics is to get a grammatical account of something like English.

How do you know when the grammar is right? Well, you compare it with the facts.

What facts? Well, the facts about what people who speak the language will tell you is grammatical and not. So even I understand that example is, in fact, a human creation, right? So that was that that's the analogy. So even even if you think or logic or mathematics as human creations, which, by the way, in some sense, they obviously are in some sense, we could have a whole different discussion there about things like, I don't know, mathematical Platonism, for instance.

But even without getting there, I think the analogy with language is apt. The language is clearly a human creation. And yet there are such things as linguistic facts that you can use to test your theories.

Well, unless I'm misunderstanding the analogy, it seems like the logical fact then would like if you ask me, I would tell you it is a logical fact that something can be both true and false at the same time.

But so how like how can you tell me about the structure of a paradox?

Doesn't actually make clear that that is the case. I mean, the liar's paradox when you analyze it does show that well, as it turns out, as illogical as a matter of logical fact that something can't be true or not true at the same time.

Well, no, no, no. That that seems like a oh, a strategy we might use to deal with this paradox, but not a strategy that is uncontroversial. Let me make a minor correction.

OK, the no contradictions are true. It's not so much a principle of logic. It's a principle of metaphysics. OK, principles of logic tend to be what follows from what. OK, so does an arbitrary conclusion full of contradiction. I mean, that's certainly a fact of logic. And when you construct a logical theory, you can test it against the facts of logic. The things that we know is this. Does this follow or does it not?

And we have intuitions about that just as we have intuitions about, you know, whether Cenizo English is grammatical. But let's come back to this question you asked right at the start about why you should suppose that contradictions can't be true. A bit of background here, generally speaking, that's been high orthodoxy in Western philosophy, not so much an Eastern philosophy, but there's only been one substantial defense of it in the history of Western philosophy. And that was by Aristotle.

And it was in the metaphysics not not the not the logical texts. The metaphysical texts. Right. And Aristotle had a bunch of arguments that have nothing to do with explosion.

But his defense of the law of no contradiction has been very, very influential because no one really has felt the need to defend the law again until recently when it's been put under pressure by people like me.

All right. So you might think that Aristotle's reasoning was very cogent. Well, it's not.

I mean, I won't go into the details here, but there's one long argument which is so tortured that experts can't even agree on what it is.

They can't even agree how it's supposed to work and that it works. And then there is six or seven small arguments, most of which are clearly beside the point. And that's, you know, that's obvious. So, you know, why it has Aristotle influence been so persuasive? Well, that's a really good sociological question.

Nearly everything else that Aristotle said has been refuted or at least seriously challenged is taken, you know, two and a half thousand years to get round to the law of no contradiction.

But has that been persuasive? Because my assumption would have been apriori, that it just is so self-evident, people, that you can have the military and at the same time that they wouldn't have needed Aristotle to think that the psychological answer there might be that it's so strongly intuitive.

But of course, the fact that if something is strongly intuitive doesn't mean that it's true. I mean, we it took us fifteen hundred years to question the idea that the earth is flat.

So not only that, when when Galileo suggested the Earth moves people to get to be obvious that he was wrong, because after all, we do feel the earth move sometimes, you know, in earthquakes and we know it's not like that. So it seemed obvious to people. So so, I mean, this seems obvious to you, but.

Well, let me ask you, why do you think that's true?

Why do I think it's true that something can't be true and false at the same time? Um, because that seems built into the definitions of true and false. To me.

The definition of true is what?

Well, I'm not going to answer that because I can't.

But but if you I guess I should just say that if if someone were to talk about something being both true and false at the same time, I would no longer know what they meant by the words true and false.

Really?

Yes. Did you not understand when we're talking about the life paradox? Yeah, I know, I understand that, I understand well, but there are some things that I know no, I wouldn't understand if something was true and false at the same time.

But but we just went through an example where you did, um. No, sorry. I meant I understood what was meant to be communicated by the liar's paradox. I don't I wouldn't understand what would be meant to be communicated by saying that the statement is both true and false at the same time.

Let let me give you an example from the history of mathematics. For most of the history of mathematics, people have thought that infinitude is actually infinitude, are incoherent because they seem to be embroiled in a bunch of paradoxes. For example, that if you take the whole numbers, note one, two, three, four, five, and you take the even numbers nought, two, four, six, eight. You compare them off one to one. So there seem to be exactly the same number of numbers and even numbers.

And that's crazy because the numbers are missing half of them. Right. So this is an argument that has been known you a long, long time by by the Greeks.

And people suppose that that showed that the whole notion of infinitude was incoherent. Mm hmm. Now, things change radically with the work of the German Danish magician George Cantor in the 19th century.

And Cantor said, no, this is what you've got to understand, is that this fact that you can throw things away from a set and have the same number of things. This is not incoherent.

This is actually the definition of infinitude.

The difference between a finite set and an infinite set is with a finite set. If you throw things away, the set gets smaller with an infinite set. If you throw things away, the set can remain the same size. And the reason you haven't got your head around this guys is because you've only been thinking of the finite, because you know, you haven't you've never taken the infinite seriously.

But now I'm going to do this. OK, so what Cantor was pointing out was that these sort of things which were paradoxical could actually be true and that they seem false just because they feel.

Well, it's what if it Kansteiner called an inadequate diet of examples, but our intuitions are trained on a set of things that doesn't say that.

So what I might suggest to you is that your intuitions about not about contradiction, not contradiction, are coming from this inadequate diet, for example, because you've only ever thought really about consistent situations. Hey, but I'm giving you some inconsistent situations and you can see that, well, you know, maybe that, like, sort of the paradox of the infinite. We just need to, you know, chill out and think about more unusual situations. There's more in heaven and earth to talk.

And you don't know.

Now, I can't let this episode finish me without mentioning something that you actually do take briefly up in in in the article, the San Francisco PD article. And that's got a lot of our listeners probably have heard. I've got ulterior theorems, actually. There's more than one incompleteness theorems. And it's I've seen an inordinate amount of misuse of those theorems, for example.

But but the reason I'm bringing it up is perhaps for one thing, for to to to ask you very briefly to actually explain what the theorem actually says that it's relevant may be relevant to, but largely because you say that a consistent approach to Arithmetic's overcomes the limitations of arithmetic that are supposed to by many to follow from Gödel's theorems.

So from what I understand you're suggesting, the consistent logic can actually be a way to sort of overcome and get around the limitations imposed on arithmetic's by Gödel's theorem. Bertrand Russell, be happy about this?

No, because you subscribe to explosion. OK. OK, let me try and take this as simply as I can.

Maybe we should start with the theorems for this year.

And indeed so girdles theorems are about formal theories of mathematics like set theory or arithmetic and so on. And what girdle showed? Well, let me state it as it's normally stated, but being very loose terms.

What girl actually showed was that provides you've got a mathematical theory which is sufficiently expressive, which you obviously want in a mathematical theory.

So nontrivial, not no, not a trivial small.

Well, that's got to contain certain mechanics, certain mathematical machinery. OK. And that's technical in nature, but I'm not going to go into that.

But if you have got a theory of this kind, then there are going to be things which are true of the topic matter, which you cannot prove in theory. OK, so the theory is specified by a set of axioms.

And what girl showed, according to the traditional story, is that the axioms are always going to be incomplete in the sense that for any axiomatic system you construct, there are going to be true things which cannot be proved. OK, so from within that system, from within us right now, actually what Girdle showed was not quite that.

What he showed was that if you've got a mathematical theory of the suit, the appropriately strong kind, it must be either incomplete or inconsistent are OK.

That's what is result actually shows. Now, of course, girl didn't know anything about Perkins's logic. And most people who write on a seems like nothing about power, consistent logic.

And so they ignore the inconsistencies. So they just didn't take the other possibility seriously.

Right.

Now, one thing that we're learning in this episode that it's all about learning things, taking things seriously.

Well, there is a lot of that going on the last few minutes. Yes. Yeah.

I mean, one thing that Parkinsonian logic does is open your eyes to all kinds of possibilities which, you know, were closed off to you before anyway.

So coming back to girls them Parkinson logic does not challenge girls result in that form.

That is the theory is either inconsistent or incomplete, but it gives you the paroxysm logic gives you the opportunity to explore the other possibility, namely that the mathematical theory is complete but inconsistent. And we now know that there are mathematical theories of this kind where you can prove everything that's true, including some contradictions. Okay, but they're important for listeners.

The gun was pointing to Julian. That raises eyebrows.

OK, so that's sort of one way of looking at what I meant when I was explaining the relevance of power consistency to girls serum's. All right.

Well, that was an impressively concise and clear explanation of incompleteness and good. All I.

I didn't expect you to be able to do that in three minutes, so. Well done. We are now out of time.

So let's wrap this up and move on to the rationally speaking PEX.

Welcome back. Every episode, we pick a suggestion for our listeners that has tickled our rational fans. This time we ask again for the second time in recent memory, our guest, Graham, pressed for his suggestion.

Okay, can I be greedy and have to turn? Okay, if they're very, very different.

And one of my favorite pieces of philosophical writing is Hume's Dialogues on Natural Religion.

Oh, we have something in common there. Okay. It's a fabulous piece of writing. It's beautifully crafted as a piece of writing.

It's philosophically engaging and cogent, and it's done in anybody in such a way that anybody can understand that this is not hard technical philosophy, human rights in a very engaging fashion.

And most of all, many of the arguments that you hear people put up nowadays for the existence of God were pretty conclusively destroyed by humor in the 18th century.

So I think if you want to talk about the philosophy of Christian religion, you need to read Hume's dialogues and that it's such a beautiful piece of writing. I'd recommend that to anybody who's interested in the existence of God and your second book.

OK, the second one, I'm going to be self-serving because what we've been talking about is logic and logic can get very technical, very fast, but it engages with a number of philosophical questions like, you know, some of them we've been discussing now. But the the engagement of logic and philosophy is manifold.

And I've always been interested in technical logic when I am because of its philosophical engagements. So some logicians are interested in proving theorems, and I do that stuff, too. But for me, if it doesn't engage with far, it's it's like playing chess, you know, it's fun, but that's it.

So I think that philosophy and logic should be taught hand in hand. And a number of years ago, I wrote a book called Logic, a very short introduction.

So Oxford University Press does this whole series of books called Very Short Introduction that the it's a really nice series.

They're meant for the kind of nut, the non-specialist in these areas.

Okay. And I've read several of these books and they're always done really well. Well, with the possible exception of mine.

Okay.

But anyway, I wrote one I wrote the book on logic in the very short introduction. And what it does is explain to people the basics of modern logic by talking about the way that these ideas engage with various philosophical problems, such as the existence of God or the nature of time or gosh, I forgot what other things are in the book but of non-existence and impossible objects and.

OK, so so I would recommend that to anybody who'd like to understand why logic is important. If you want to think about some issues in philosophy, I will.

Second, that I have read the book, so I haven't read it, but now I'm dying to ask.

Graham, thank you so much for joining us again on rationally speaking, it's been quite a pleasure having you meet jury. This concludes another episode of rationally speaking. Join us next time for more explanations on the borderlands between reason and nonsense.

The rationally speaking podcast is presented by New York City skeptics for program notes, links, and to get involved in an online conversation about this and other episodes, please visit rationally speaking podcast Dog. This podcast is produced by Benny Tollan and recorded in the heart of Greenwich Village, New York. Our theme, Truth by Todd Rundgren, is used by permission. Thank you for listening.